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{\large Complexity in low-dimensional topology}
Let $\cal O$ be a class of geometric objects. By a {\em complexity
function} on $\cal O$ we mean a map $c\colon O\to {\Bbb N}\cup \{
0\}$. Of course, we expect that $c$ is natural, i.e. it measures
how complicated a combinatorial description of a given object must
be. Using $c$, we can potentially do the following:
\begin{enumerate}
\item Convert observations into rigorous statements.
\item Classify objects of small complexity.
\item Use inductive arguments.
\end{enumerate}
There are several ways to construct reasonable complexity
functions. First, one can try to count the number of
singularities of a given object $A\in \cal O$. For example, a
natural complexity of a graph will probably contain information on
the number of its vertices. But what to do if the objects in
question are homogenous, like manifolds? Then one can count
singularities of different presentations of a given object and
take the minimum. For example, knots are usually tabulated
according to the number of double points of their minimal
projections.
Based on this idea, we introduce
different complexity functions for 3-manifolds,
compare them, and describe
their properties. One of them ({\em spine complexity})
is very close to the number of tetrahedra in a minimal
triangulation, the other ({\em Heegaard complexity}) takes into
account the number of crossing points of the meridians that form
a minimal Heegaard diagram. As computer experiments show, the
number of 3-manifolds having complexity $n$ grows exponentially
in the first case and only polynomially in the second one.
We introduce also an {\em extended complexity}, which is a
universal tool for inductive proofs.
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